Germany’s announcement of extra funds came after talks between the leaders of Chancellor Merkel’s coalition.

In addition to the €3bn ($3.3bn; £2.2bn) for the regions, they also agreed to speed up asylum application procedures, provide extra accommodation for newcomers, and allocate funds to pay benefits to new arrivals.

Germany expects to receive 800,000 refugees and migrants this year, and wants to see the rest of Europe do more to help.

But while Ms Merkel has become a hero to many migrants and their supporters, conservative allies said she sent a “totally wrong signal” by allowing in the intake from Hungary.

The interior ministry said the decision was an exception to help avert a humanitarian crisis.

New arrivals in Germany were welcomed by smiling and cheering members of the public at train stations across the country.

On Sunday, a group of cars driven by German and Austrian activists travelled to the Hungarian border to pick up migrants and distribute food.

The migrants had travelled north through the Balkans – Greece, Macedonia and Serbia – before arriving at Hungary’s southern border, and on to Austria and Germany.

Syrians are the largest group travelling, followed by Afghans and Eritreans.

A rift has developed within the EU over how to deal with the crisis.

Hungary has accused Germany of encouraging the influx, and is pressing ahead with plans to tighten border controls and could send troops to its southern frontier if parliament agrees.

It has opened a new reception camp for migrants in southern border village of Roszke, and is due to finish a border fence this month.

The UN’s Refugee Chief Antonio Guterres said the crisis was “manageable” if member states could agree a joint plan.

A note on terminology: The BBC uses the term migrant to refer to all people on the move who have yet to complete the legal process of claiming asylum. This group includes people fleeing war-torn countries such as Syria, who are likely to be granted refugee status, as well as people who are seeking jobs and better lives, who governments are likely to rule are economic migrants.

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